THE MOTION OF A PROJECTILE 825 



- a? cos x + 2 {x sin x Jsin x dx] -r- c 



- a? 2 cos x + 2a? sin x + 2 cos x + c 

 = (2 a; 2 ) cos x + 2x sin x + c 



167. The Motion of a Projectile. One law of air resistance is 

 that, if R Ib. is the resistance, d ft. is the diameter of the projectile, 

 and v ft. per second is the horizontal component of the velocity 

 at any instant. 



Then R = 2d*(v - 850) 



If m Ib. is the mass of the projectile, 



R = x acceleration 



g 



2d?(v -850)= - 



dt m 



dv 



- 850 m 



lg e ( v ~ 85 ) = ^ + C 



m 



Let z> ft. per second be the initial horizontal muzzle velocity 

 Then, when t = 0, v = v , and log^o 850) = C. 



Hence log e (i> - 850) - log e (i; - 850) = - - t 



llv 



v - 850 2gd? A 



b 



= e 



- 850 



- 

 and ^ z; = 850 +(v - 850)e m 



ds 

 -3- 



ds 

 This gives the velocity at any instant, but v = -3- 



ds 



-- = 850 + (y - 850)^ 



Integrating, * = 850* - ~ e~ + 



